12. Insert 5 geometric means between 35 and 40§.

20. The sum of the first 6 terms of a G. P. is 9 times the sum of the first 3 terms; find the common ratio.

21. The fifth term of a G. P. is 81, and the second term is 24; find the series.

22. The sum of a G. P. whose common ratio is 3 is 728, and the last term is 486; find the first term.

23. In a G. P. the first term is 7, the last term 448, and the sum 889; find the common ratio.

24. The sum of three numbers in G. P. is 38, and their product is 1728; find them.

25. The continued product of three numbers in G. P. is 216, and the sum of the product of them in pairs is 156; find the numbers.

26. If S, denote the sum of the series 1++2+... ad inf., and s, the sum of the series 1−7+72p – ad inf., prove that

that

Sp+&p=2S2p

27. If the pth, qth, 7th terms of a G. P. be a, b, c respectively, prove aq-r br-p cp-q=1.

28. The sum of an infinite number of terms of a G. P. is 4, and the sum of their cubes is 192; find the series.

58. Recurring decimals furnish a good illustration of infinite Geometrical Progressions.

which agrees with the value found by the usual arithmetical rule.

59. The general rule for reducing any recurring decimal to a vulgar fraction may be proved by the method employed in the last example; but it is easier to proceed as follows.

To find the value of a recurring decimal.

Let P denote the figures which do not recur, and suppose them p in number; let Q denote the recurring period consisting of q figures; let D denote the value of the recurring decimal; then D= PQQQ...........;

and

... 10o × D= P·QQQ ........

10p+q x D = PQ·QQQ .....

.......

;

therefore, by subtraction, (10P+ – 10o) D = PQ − P ;

Now 10-1 is a number consisting of q nines; therefore the denominator consists of q nines followed by p ciphers. Hence we have the following rule for reducing a recurring decimal to a vulgar fraction:

For the numerator subtract the integral number consisting of the non-recurring figures from the integral number consisting of the non-recurring and recurring figures; for the denominator take a number consisting of as many nines as there are recurring figures followed by as many ciphers as there are non-recurring figures.

60. To find the sum of n terms of the series

a, (a+d)r, (a + 2d) r2, (a + 3d) r3,

in which each term is the product of corresponding terms in an arithmetic and geometric series.

Denote the sum by S; then

S= a + (a + d) r + (a + 2d) y2 + ...

.. rs =

+ (a + n − 1d) r"';

ar + (a + d) r2 + ... + (a + n − 2d) r"-1+(a+n−1d)r".

By subtraction,

S (1 − r) = a + (dr + dr2 + ... + dr"-') − (a + n − 1d) r"

then if r<1, we can make " as small as we please by taking n sufficiently great. In this case, assuming that all the terms which involve rn can be made so small that they may be neglected, we

to this point again in Chap. XXI.

In summing to infinity series of this class it is usually best to proceed as in the following example.

3. Sum 1+3x+5x2+7x3+9x1+... to infinity.

7. Prove that the (n + 1)th term of a G. P., of which the first term is a and the third term b, is equal to the (2n+1)th term of a G. P. of which the first term is a and the fifth term b.

8. The sum of 2n terms of a G. P. whose first term is a and common ratior is equal to the sum of n of a G. P. whose first term is b and common ratio 2. Prove that b is equal to the sum of the first two terms of the first series.

9. Find the sum of the infinite series

1+(1+b) r+(1+b+b2) r2+(1+b+b2+b3) μ3 + ...

r and b being proper fractions.

10. The sum of three numbers in G. P. is 70; if the two extremes be multiplied each by 4, and the mean by 5, the products are in A. P.; find the numbers.

11. The first two terms of an infinite G. P. are together equal to 5, and every term is 3 times the sum of all the terms that follow it; find the series.

Sum the following series :

12. x+α, x2+2a, x3+3a... to n terms.

13. x(x+y)+x2 (x2 + y2)+x3 (x13+y3)+... to n terms.

18. If the arithmetic mean between a and b is twice as great as the geometric mean, shew that a: b=2+√3 : 2√3.

19. Find the sum of n terms of the series the 7th term of which is

(2r+1) 2r.

20. Find the sum of 2n terms of a series of which every even term is a times the term before it, and every odd term c times the term before it, the first term being unity.

21. If S, denote the sum of n terms of a G. P. whose first term is a, and common ratio r, find the sum of S1, S3, S5,...S2n-1.

22. If S1, S2, S3,...S, are the sums of infinite geometric series, whose first terms are 1, 2, 3,...p, and whose common ratios are

23. If r < 1 and positive, and m is a positive integer, shew that

Tence shew that nm is indefinitely small when n is indefinitely great.